I'm probably not the best person to answer this, but I'll try. Basically three-act structure is the most common form of a story and has been around since the ancient Greece. Act 1 introduces the characters and the setting. Act 2 sets some kind of obstacle, problem or opposing force, which can be anything from trying to win a sports event to a zombie epidemic. Many times there's some kind of antagonist(s). In act 3 the problem gets solved and the characters will have mentally "grown" in some way. Usually the protagonist(s) are at their lowest point just before the third act, and things seem the most desperate just before they get better.

And just in case you don't know, protagonist = main character, antagonist = bad guy.

Examples of "non-three-act-structured" movies would include titles such as 2001: Space Odyssey, Pulp Fiction, Magnolia, Lost in Translation, Fear and Loathing in Las Vegas, most films by David Lynch, etc...